Hook Multiplication in the Quantum K-Theory of Grassmannians

TL;DR

This work analyzes the quantum K-theory ring $QK(Gr(m,n))$ for Grassmannians and proves a manifestly positive formula for hook multiplication: the product $\mathcal{O}^{\lambda}\mathcal{O}^{(a\backslash b)}$ has a distinguished maximal term $q\mathcal{O}^{\lambda}$ with coefficient $N_{\lambda,(a\backslash b)}^{\lambda[1]}$, computable from classical $K$-theory structure constants. The authors develop the quantum poset framework, prove a hook-multiplication formula, and show a reduction to a universal case $c(t,a,b)$ determined by the number of quantum corners $t$ of $\lambda$, via Theorems main0–main2. They also provide a combinatorial interpretation of $c(t,a,b)$ through set-valued tableaux and relate it to the classical Littlewood-Richardson rule after translating shapes to the staircase $\rho_t$, yielding a bridge between classical and quantum constants. The results generalize the quantum Pieri rules and give a positive, interpretable combinatorial picture for hook multiplication in $QK(Gr(m,n))$, with potential impact on computations and understanding of quantum $K$-theoretic structure constants.

Abstract

We study the quantum K-theory ring $QK(X)$ of a Grassmannian $X$ and prove a manifestly positive formula for the product of an arbitrary class by a hook class. This generalizes the quantum K-theoretic Pieri rule, a prior result of Buch and Mihalcea. We also present a combinatorial interpretation of this result.

Figures (8)

• Figure 1: On the left, we see the hook shape $(3\backslash2)$, and on the right, we see the staircase shape $\rho_5$.
• Figure 2: Each of these diagrams has 3 quantum corners. On the left, ${\lambda}_1=(3,3,2,0)$ has 2 corners and ${\lambda}_1^\vee=(4,2,1,1)$ has 3 corners. In the middle, ${\lambda}_2=(4,3,2,0)$ has 3 corners, as does ${\lambda}_2^\vee=(4,2,1,0)$. On the right, ${\lambda}_3=(4,2,2,1)$ has 3 corners and ${\lambda}_3^\vee=(3,2,2,0)$ has 2 corners.
• Figure 3: These are two diagrams of ${\lambda}=(5,5,3,0)$ in ${\mathbb Z}^2/{\mathbb Z}(4,-6)$. In both, the heavy line is the southeastern boundary of ${\lambda}$. On the left we only represent boxes with coordinates from $(1,1)$ to $(4,6)$, and on the right we show a portion of the corresponding order ideal in ${\mathbb Z}^2$.
• Figure 4: The shape $\rho_5$ as an element of ${\mathbb Z}^2/{\mathbb Z}(5,-5)$. It has 4 corners when considered as a classical shape, but 5 quantum corners.
• Figure 5: We consider the setup in Remark \ref{['rem:reduction']} where $m=5$, $n=9$, ${\lambda}=(3,3,2)$, $t=3$, $a=4$, $b=2$. The diagrams on the left show successive stages of removing a repeated row or column from ${\lambda}$, and the diagrams on the right correspondingly remove a row or column from $(a\backslash b)$. This process terminates when ${\lambda}$ has become $\rho_3$ and $(a\backslash b)$ has become $(a-m+t,b-n+m+t)$.

Theorems & Definitions (32)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4: Lenart_option1, buch_qkgrass, buch_qkpieri
• Lemma 5
• proof
• Corollary 6
• proof
• Lemma 7
• proof